The prototypical problem we study here is the following. Given a 2L x 2L square, there are approximately \exp(4KL^2/\pi) ways to tile it with dominos, i.e. with horizontal or vertical 2 x 1 rectangles, where K\approx 0.916 is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. An algorithmically simple way of sampling uniformly one among so many tilings is to introduce a Markov Chain algorithm where, with rate 1, two adjacent horizontal dominos are flipped to vertical dominos, or... Source: http://arxiv.org/abs/1210.5456v1

The paper deals with the problem of finding the best alternatives on the basis of pairwise comparisons when these comparisons need not be transitive. In this setting, we study a reinforcement urn model. We prove convergence to the optimal solution when reinforcement of a winning alternative occurs each time after considering three random alternatives. The simpler process, which reinforces the winner of a random pair does not always converges: it may cycle. Source: http://arxiv.org/abs/1301.5734v1

byNathanaël Berestycki; Benoit Laslier; Gourab Ray

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We present a general result which shows that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds true assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. This includes the case of lozenge... Topics: Probability, Mathematical Physics, Mathematics Source: http://arxiv.org/abs/1603.09740

byNathanaël Berestycki; Benoit Laslier; Gourab Ray

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The purpose of this note is to give a succinct summary of some basic properties of T-graphs which arise in the study of the dimer model. We focus in particular on the relation between the dimer model on the heaxgonal lattice with a given slope, and the behaviour of the uniform spanning tree on the associated T-graph. Together with the main result of the companion paper \cite{BLR16}, the results here show Gaussian free field fluctuations for the height function in some dimer models. Topics: Probability, Combinatorics, Mathematics Source: http://arxiv.org/abs/1610.07994

byNathanaël Berestycki; Benoît Laslier; Gourab Ray

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In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8}. $$ where $\kappa' $ is the SLE parameter associated with this model.... Topics: Mathematics, Probability, Mathematical Physics Source: http://arxiv.org/abs/1502.00450